Second cohomology space of $\mathfrak{sl}(2)$ acting on the space of bilinear bidifferential operators

We consider the sl(2)-module structure on the spaces of bilinear bidifferential operators acting on the spaces of weighted densities. We compute the second cohomology group of the Lie algebra sl(2) with coefficients in the space of bilinear bidifferential operators that act on tensor densities Dλ,ν,µ.